Permutation-Invariant Quantum Codes for Deletion Errors

Abstract

This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes give the first example of quantum codes that can correct two or more deletion errors. Also, our codes give the first example of quantum codes that can correct both multiple-qubit errors and multiple-deletion errors. We also discuss a generalization of the construction of our codes at the end.

The equivalence between correctability of deletions and insertions of separable states in quantum codes

“…A systematic construction of single deletion codes that encompasses these examples has been proposed [13], with more examples given by Shibayama [21]. Since it is clear that erasure and deletion errors are equivalent in permutation-invariant codes [16,22], Ouyang's permutation-invariant quantum codes [14,15] are also quantum deletion codes. Research on quantum codes correcting insertion errors on the other hand has only just begun, with the discovery that the four qubit deletion code can also correct a single insertion error [5].…”

Constructions of $\ell$-Adic $t$-Deletion-Correcting Quantum Codes

“…Those researches provided concrete examples of quantum deletion-correcting codes. The first systematic construction of 1deletion-correcting binary quantum codes was proposed in [10], where ((2 k+2 − 4, k)) 2 codes were constructed for any positive integer k. Very recently, the first systematic construction of t-deletioncorrecting binary quantum codes was proposed [20,19] for any positive integer t. The number of codewords was two in [20,19]. There are the following problems in the existing studies: (1) There is no systematic construction for nonbinary quantum codes correcting more than 1 deletions.…”